The Paisley Leopard

The helical shell of a snail, the hexagonal chambers of a honeycomb, the tiger’s stripes and the leopard’s spots, leaf shapes in their spectacular variety, the sixfold symmetry of a snowflake, the branching pattern of an elm tree or a river delta or a stag’s antlers, the curiously similar profiles of ocean waves and sand dunes—you might think that explaining these natural forms would always have been at the core of the scientific enterprise. Yet, on the contrary, the science of shape, pattern and form is a modern preoccupation. Earlier observers described and catalogued many such features of the natural world, but only in the past century or so has it seemed worthwhile to search for the principles and mechanisms—and ultimately the algorithms—that give rise to those patterns.

Philip Ball, in a trilogy of books published under the general title Nature’s Patterns, offers some thoughtful commentary on why the science of morphology has been such a late bloomer. He notes that making things requires skill and attention to detail; in everyday experience, complex objects don’t just assemble themselves spontaneously. In Shapes he observes that

patterning the world, shaping it into forms that please us or do useful things, is hard work. . . .

So when the natural philosophers of ages past found complexity in nature, it is scarcely surprising that many of them decided they were gazing on God’s handiwork and artistry.

If every atom is put in its place as part of a grand design, then delving into the principles of that design is the work of theology, not science. Indeed, the Reverend William Paley, the great exponent of this view, titled his 1802 treatise on the subject Natural Theology.

Darwinism brought a new worldview, yet it still offered little insight into the mechanism of pattern. The deistic dictum “God gave the leopard spots to help it hunt” was restated as “Spotted leopards leave more progeny than unspotted ones,” but this new formulation answered only the why question, not the how question. Mutation and natural selection were treated as omnipotent forces that might as easily produce a paisley or a plaid leopard if those patterns offered an adaptive advantage. The whole matter of how an organism would paint itself with spots (or paisley swirls, for that matter) was left as a problem for someone else to solve.

The someone else who first gave serious attention to questions of this kind was the Scottish naturalist D’Arcy Wentworth Thompson, whose eye-opening book On Growth and Form first appeared in 1917. Thompson imported ideas from physics and mathematics and even engineering into biology, including the aesthetic notion that elegance and economy of means are important virtues in a scientific theory. Ball describes Thompson’s analysis of the shapes of animal horns:

The sabre-like sweep of an ibex horn need not have been selected from a presumed gallery of bizarre and ornate alternative horn shapes. We can assume merely that the horn grows at a progressively slower rate from one side of the circumference to the other, whereupon, hey presto, you have an arc. . . . Even the more elaborate spiral form of a ram’s horn comes simply from ramping up the asymmetry of growth rates, causing the horn’s tip to swing through several complete revolutions.

D’Arcy Thompson did not directly address the issue of the leopard’s spots, but Alan M. Turing did, in the early 1950s. He invented a scheme based on two competing chemical species—an activator and an inhibitor—that can generate a wide variety of pelt patterns, from spots to stripes to the irregular blotches of a Holstein cow. Converting from one pattern to another is just a matter of adjusting a few parameters, such as the rate at which the chemicals diffuse through a layer of tissue.

By now the idea of self-organizing patterns is no longer a novelty. They seem to be everywhere, not only in living organisms but also in the inanimate world. Ball’s trilogy is a grand tour of these systems. Shapes provides a general introduction (including discussions of both Thompson and Turing) and a wide-ranging survey of patterns observed in soap bubbles and foams, in the skeletons of marine microorganisms, in mollusk shells and butterfly wings, in oscillating chemical reactions, in the spiral patterns of flowers, in population cycles among predators and their prey, and in embryonic development. Flow, the second volume, takes on not only vortices, convection and turbulence but also some less-obvious topics such as the motion of sand dunes and the self-organizing behavior of flocks and herds of animals and even highway traffic. Finally Branches includes the forms of trees (and, at a smaller scale, the veins of their leaves) as well as fracture patterns, drainage networks and a few social and technological artifacts such as the electric-power grid and the Internet. This last volume ends with a useful epilogue that sums up the subject and returns to the question of why a science of pattern came to flourish only in the second half of the 20th century.

The common thread binding all the topics together is the idea of orderly patterns that form with no need for central planning or direction. For example, floating bubbles tend to assemble in rafts that approximate a hexagonal tiling of the plane, but there is no agency enforcing this regular geometry, and no hidden template or other element of the system with distinctive sixfold symmetry. The pattern arises from nothing but attractive forces between neighboring bubbles. Other hexagonal patterns turn up in the cracked mud of a dry lake bed, in fluid convection cells and in honeycombs and wasp nests; each of these phenomena has a slightly different physical cause, but in all cases the pattern always forms spontaneously. A tiled bathroom floor is quite different: You can’t rely on the tiles to arrange themselves in the proper order.

I can’t resist observing that words, too, seldom assemble themselves into coherent sentences and paragraphs; they require the single-minded attention of a writer to set them in order. Philip Ball is a very skilled practitioner of this art. He’s an explainer. In these volumes he undertakes the particularly delicate task of explaining mathematical subjects to an audience assumed to have no mathematical knowledge. He succeeds.

Even readers who are already familiar with the main ideas in these books are sure to find unexpected pleasures. My personal favorite is an extended meditation on Leonardo da Vinci’s depictions of water, which introduces the volume on fluids. Ball writes:

While most painters used technique to create a simulacrum of nature, Leonardo felt that one could not imbue the picture with life until one understood how nature does it. His sketches, then, are not exactly studies but something between an experiment and a diagram—attempts to intuit the forces at play.

A couple of pages later, he adds that

To judge from his sketches, Leonardo conducted a thorough, if haphazard, experimental programme on the flow patterns of water, watching it pass down channels of different shapes, charting the chaos of plunging waterfalls, and placing obstacles in the flow to see how they generated new forms.

The drawings that record the outcome of these experiments feature a graphic device now known to fluid dynamicists as the streamline—an attempt to capture motion in a still drawing. Leonardo’s streamlines resemble the braids and curls of flowing hair, and indeed he adopted similar stylistic elements when he was drawing hair (including his own beard in self-portraits). Later artists in the European tradition favored a less dynamic view of turbulent flow; Ball describes the typical seascape of the 18th or 19th century as “a play of glinting highlights and surging foam: a style that is all surface, you might say.” In China and Japan, however, drawings of fluid streamlines have a heritage that goes back even further than Leonardo. Ball notes that the streamlines are meant “to portray the inner life of flow.” Perhaps the concept is not so far from the inner logic of a pattern.

The three volumes of Nature’s Patterns are based on an earlier one-volume work, The Self-made Tapestry: Pattern Formation in Nature (2001). The expanded and updated edition is welcome, but I’m not sure that the trisection of the text was a good idea. For reading on the bus or the beach, I suppose, the new smaller volumes are handier than the single compendium of the old edition, but in total the three new books are bulkier and more costly. Furthermore, many of the photographs and drawings are not reproduced as well as they were in the earlier edition. This is a disappointment, since the discussion of pattern relies heavily on visual illustration. The murky printing afflicts only the black-and-white illustrations within the text; each volume also includes a few pages of color plates. And the words are still crystal clear.

Brian Hayes is Senior Writer for American Scientist. He is the author most recently of Group Theory in the Bedroom, and Other Mathematical Diversions (Hill and Wang, 2008).